ROBERT POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE, HON. LL.D., WILLIAM AND MARY COLLEGE, VA., U.S. EUCLID'S ELEMENTS OF GEOMETRY. 1. Euclid's Elements of Geometry, the University Edition, with Notes, Questions, and Geometrical Exercises, selected from the Cambridge Senate House and College Examination Papers, with Hints for Solution of the Exercises. Demy 8vo., pp. 520, 10s. 2. The School Edition, with Notes, Geometrical Exercises, &c. 12mo., pp. 418, 4s. 6d. The School Edition has also been published in the following portions, with the Notes, &c., to each book: The University Edition of Euclid's Elements was first published in 1845, and the first School Edition in 1846. Both Editions have been enlarged and improved from time to time, and the total sales of copies of the work up to the present year amount to a number very considerably above half-a-million. In the year 1853, the Council of Education at Calcutta were pleased to order the introduction of these Editions of Euclid's Elements into the Schools and Colleges under their control in Bengal. In the year 1860, a Translation of the Geometrical Exercises was. made into the German Language, by Hans H. Von Aller, with a Preface by Dr. Wittstein, and published at Hanover. At the International Exhibition of 1862, in London, a Medal was awarded to R. Potts, "For the Excellence of his Works on Geometry.' Jury Awards, Class XXIX., p. 313. "In my opinion Mr. Potts has made a valuable addition to Geometrical literature by his Editions of Euclid's Elements."-W. Whewell, D.D., Master of Trinity College, Cambridge. (1848.) "Mr. Potts has done great service by his published works in promoting the study of Geometrical Science."-II. Philpott, DD., Master of St. Catharine's College. (1848.) "Mr. Potts' Editions of Euclid's Geometry are characterized by a due appreciation of the spirit and exactness of the Greek Geometry, and an acquaintance with its history, as well as by a knowledge of the modern extensions of the Science. The Elements are given in such a form as to preserve entirely the spirit of the ancient reasoning, and having been extensively used in Colleges and Public Schools, cannot fail to have the effect of keeping up the study of Geometry in its original purity."-J, Challis, M.A., Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridge. (1848.) "Mr. Potts' Edition of Euclid is very generally used in both our Universities and in our Public Schools; the notes which are appended to it shew great research, and are admirably calculated to introduce a student to a thorough knowledge of Geometrical principles and methods."-George Peacock, D.D., Lowndean Professor of Mathematics in the University of Cambridge, and Dean of Ely. (1848) "By the publication of these works, Mr. Potts has done very great service to the cause of Geometrical Science. I have adopted Mr. Potts' work as the text-book for my own Lectures in Geometry, and I believe that it is recommended by all the Mathematical Tutors and Professors in this University."-R. Walker, M.A., F.R.S., Reader in Experimental Philosophy in the University, and Tutor of Wadham College, Oxford. (1848.) LONDON: LONGMANS & CO., PATERNOSTER ROW. ROBERT POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE. HON. LL.D. WILLIAM AND MARY COLLEGE, VA., U.S. PALEY'S EVIDENCES OF CHRISTIANITY and the Horse Paulinæ, edited with Notes; with an Analysis. and a selection of Examination Questions from the Cambridge Papers. 8vo., pp. 588, 10s. 6d., cloth. "Mr. Potts' is the most complete and useful edition yet published."-Eclectic Review. We feel that this ought to be henceforth the Standard Edition of the Evidences and the Hora."-Biblical Review. "The scope and contents of this new edition of Paley are pretty well expressed in the title. The Analysis is intended as a guide to Students not accustomed to abstract their reading, as well as an assistance to the mastery of Paley; the Notes consist of original passages referred to in the text, with illustrative observations by the Editor the questions have been selected from the examinations for the last thirty years."-Spectator. A BRIEF ACCOUNT OF THE Open to competition in the University of Cambridge, with Specimens of the Examination Papers. Feap. 8vo, pp. 157, cloth, 1s. 6d. LIBER CANTABRIGIENSIS, An account of the Aids, Encouragements, and Rewards open to Students in the University of Cambridge. Fcap. 8vo., pp. 570, bds., 4s. 6d. MAXIMS, APHORISMS, &c., FOR LEARNERS. Double crown, bds., pp. 192, 1s. 6d. LONDON: LONGMANS & CO., PATERNOSTER ROW. KING EDWARD VI. ON THE SUPREMACY, With an English Translation, and a few brief notices of his Life, Education, and Death. Double crown, cloth bds., gilt edges, 2s. 6d. This short treatise is printed from the autograph copy of King Edward VI., preserved in the Cambridge University Library, and is really a literary curiosity, whether it be regarded in reference to the author or the subject. CAMBRIDGE: W. METCALFE & SON. LONDON: N. S. DEPOSITORY. A CHAPTER OF ENGLISH HISTORY ON THE SUPREMACY OF THE CROWN, With an Appendix of Public Documents. Svo. CAMBRIDGE: W. METCALFE & SON. ELEMENTARY ARITHMETIC, WITH BRIEF NOTICES OF ITS HISTORY. SECTION XII. LOGARITHMS. BY ROBERT POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE, HON. LL.D. WILLIAM AND MARY COLLEGE, VA., U.S. CAMBRIDGE: PUBLISHED BY W. METCALFE AND SON, TRINITY STREET. LONDON: SOLD AT THE NATIONAL SOCIETY'S DEPOSITORY, WESTMINSTER, 1876. W. METCALFE AND SON, TRINITY STREET, CAMBRIDGE. ...6d. NOTICE. As the Book-post affords great convenience for the prompt transmission THE PROPERTIES AND CONSTRUCTION OF LOGARITHMS. u= ART. 1. DEF. 1. In the equation u = a, where a is a constant number greater than unity, and u any natural number, the index x is defined to be the logarithm of the number u to the base a. The notation assumed to denote "the logarithm of the number u to the base a" is logu, so that x=log", and the equation u=a* may be written u=alogau ̧ DEF. 2. The base of any system of logarithms is any fixed number which, being raised to the powers denoted by the logarithms, produces the successive natural numbers. DEF. 3. A system of logarithms is a series of the successive values of a derived from the equation u=a", when the natural numbers 0, 1, 2, 3, 4, &c., are successively substituted for u, the same base a being preserved.1 1 Logarithms may be defined to be, as in fact they are, a series of numbers in Arithmetical progression which increase by a common difference, corresponding to another series in Geometrical progression which increase by a common multiplier. For example, let 10 be made the base, Then the numbers 0, 1, 2, 3, 4, &c., are the logarithms of the series of numbers 1, 10, 100, 1000, 10000 &c., respectively, to the base 10. Hence it is obvious that a negative number cannot be assumed as the base of any system of logarithms; for the odd powers of a negative number are negative, and the even powers are positive, and consequently they are not subject to the law of continuity in producing in order all the natural numbers. This definition of a system of logarithms suggests a method of finding the logarithms of all the intermediate numbers; for the Arithmetic mean between any two consecutive terms of the Arithmetic series will be the logarithm of the Geometric mean of the two corresponding terms of the Geometric series : Again the A. mean between 1 and 2 is 1·5, and the G. mean between 10 and 100 is 31.6227766; and so on for successive mean proportionals. Next, the A. means can be found between every two consecutive terms of the A. series 0, 5, 1, 1·5, &c., and the G. means between every two corresponding terms of the G. series 1, 3.1622777, 10, 31 6227766, 100, &c.; and so continuing the same process, may be found the logarithms of all numbers, but at very great expense of time and labour. |